Mathematical Sciences: C*-Algebras and the Geometry of Hilbert Space

这是一项数学研究，旨在解决两个著名的开放问题，它们与算子代数上的纯态有关。状态这个词来自量子力学。在某些表述中，可观测量用适当的算子代数中的自伴随算子表示，状态是算子代数上的某些线性泛函；因此，指定一个状态为每个可观察对象分配一个数值。纯状态是那些没有以其他状态的凸组合形式给出的状态。原则上，它们决定了代数的一切，但更精确地确定它们之间的联系却带来了令人生畏的困难。本项目将解决的问题之一是将Stone-Weierstrass定理扩展到一般的非交换情况，Stone-Weierstrass定理可以被认为是关于交换算子代数的结果。另一个问题是证明Hilbert空间上所有有界算子代数的极大对角子代数的纯态扩展的唯一性。

This is mathematical research in pursuit of two famous open problems having to do with pure states on operator algebras. The term state comes from quantum mechanics. In some formulations thereof, observables are represented by selfadjoint operators in an appropriate algebra of operators, and states are certain linear functionals on the operator algebra; thus, specifying a state assigns a numerical value to each of the observables. The pure states are those not given as convex combinations of other states. In principal, they determine everything about the algebra, but pinning down the connection more precisely has led to daunting difficulties. One of the problems the present project will address is that of extending the Stone-Weierstrass theorem, which may be thought of as a result about commutative operator algebras, to the general, noncommutative situation. The other problem is to show the uniqueness of pure state extensions from maximal diagonal subalgebras of the algebra of all bounded operators on a Hilbert space.